|
| |
| |
|
|
|
1, 3, 15, 255, 65535, 4294967295, 18446744073709551615, 340282366920938463463374607431768211455, 115792089237316195423570985008687907853269984665640564039457584007913129639935
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,2
|
|
|
COMMENTS
|
In a tree with binary nodes (0, 1 children only), the maximum number of unique child nodes at level n.
Number of binary trees (each vertex has 0, or 1 left, or 1 right, or 2 children) such that all leaves are at level n. Example: a(1) = 3 because we have (i) root with a left child, (ii) root with a right child and (iii) root with two children. a(n)=A000215(n)-2. - Emeric Deutsch, Jan 20 2004
Similarly, this is also the number of full balanced binary trees of height n. (There is an obvious 1-to-1 correspondence between the two sets of trees.) - David Hobby (hobbyd(AT)newpaltz.edu), May 02 2010
Partial products of A000215.
The first 5 terms n (only) have the property that phi(n)=(n+1)/2, where phi(n)=A000010(n) is Euler's totient function. - Lekraj Beedassy, Feb 12 2007
If A003558(n) is of the form 2^n and A179480(n+1) is even, then (2^(A003558(n) - 1) is in the set A051179. Example: A003558(25) = 8 with A179480(25) = 4, even. Then (2^8 - 1) = 255. - Gary W. Adamson, Aug 20 2012
|
|
|
REFERENCES
|
M. Aigner and G. M. Ziegler, Proofs from The Book, Springer-Verlag, Berlin, 1999; see p. 4.
|
|
|
LINKS
|
Vincenzo Librandi, Table of n, a(n) for n = 0..13
For rate of growth see A. V. Aho and N. J. A. Sloane, Some doubly exponential sequences, Fib. Quart., 11 (1973), 429-437.
Index entries for sequences of form a(n+1)=a(n)^2 + ...
|
|
|
FORMULA
|
a(n)=A000215(n)-2.
a(n) = (a(n-1) + 1)^2 - 1, a(0) = 1. [ or a(n) = a(n-1)(a(n-1) + 2) ].
1 = 2/3 + 4/15 + 16/255 + 256/65535...= Sum(n>=0, A001146(n)/a(n+1) ); with partial sums: 2/3, 14/15, 254/255, 65534/65535... - Gary W. Adamson, Jun 15 2003
a(n)=b(n-1) where b(1)=1, b(n) = prod(k=1..n-1, b(k)+2 ) - Benoit Cloitre, Sep 13 2003
A136308(n) = A007088(a(n)). - Jason Kimberley, Dec 19 2012
|
|
|
MATHEMATICA
|
Table[2^(2^n)-1, {n, 0, 9}]..and/or..a=1; lst={a}; Do[b=(a+2)*a; AppendTo[lst, b]; a=b, {n, 1, 9}]; lst [Vladimir Joseph Stephan Orlovsky, Mar 16 2010]
|
|
|
PROG
|
(PARI) a(n)=if(n<0, 0, 2^2^n-1)
(MAGMA) [2^(2^n)-1: n in [0..8]]; // Vincenzo Librandi, Jun 20 2011
|
|
|
CROSSREFS
|
Cf. A001146, A007018.
Cf. A003558, A179480
Sequence in context: A139289 A116518 A050474 * A122591 A120607 A013352
Adjacent sequences: A051176 A051177 A051178 * A051180 A051181 A051182
|
|
|
KEYWORD
|
nonn,easy,nice
|
|
|
AUTHOR
|
Alan DeKok (aland(AT)ox.org)
|
|
|
STATUS
|
approved
|
| |
|
|
